Anders Linnér

Some properties of the curve straightening flow in the plane

We will explicitly compute the gradient of the total squared curvature functional on a space of closed curves. An example shows that the flow along the gradient trajectory may cause curves to develop self-intersections. We prove the existence of strictly convex curves that momentarily turn nonconvex. In conclusion we use computer graphics to illustrate how self-intersections come about.

Explicit Elastic Curves

The equilibria of thin rods are given by curves which are critical points of the modified total squared curvature. The critical curves are known as elastic curves. It is shown how all the elastic curves are given explicitly in terms of elliptic functions as soon as a certain set of three parameters is known. Every regular curve can be parametrized to have a constant speed but the parametrization is rarely known explicitly. Remarkably, all the elastic curves are here explicitly parametrized to have a constant speed. Curves with fixed distinct endpoints as well as closed curves are admitted. The tangent direction may be constrained at one, both, or neither of the endpoints. There are three major strands of formulas corresponding to: fixed length L, variable length without tension, and variable length with tension (let ν > 0 and add a term νL to the total squared curvature). In the most complicated cases the three parameters are given as solutions to a non-linear system of three equations. In the least complicated case everything is given explicitly in terms of elliptic functions. If the length is variable and there is no tension, at least one of the parameters is completely determined (the elliptic modulus m = 1/2). Using the same set of parameters explicit formulas are given for: the length when it is variable, the total squared curvature, and the tangent angle along the elastic curve. A number of examples are presented which illustrate the full range of constraints.

Existence of free nonclosed Euler-Bernoulli elastica

THE TOTAL squared curvature functional, F(y) = S, k2 ds, has appeared in many different contexts. In differential geometry, F has emerged in conjunction with the development of flow methods used to locate closed geodesics, see [l-S]. Since geodesics are global minima of F, it is natural to attempt to introduce a Hilbert manifold of curves in which the negative gradient flow may be used to locate the geodesics.

A unique graph of minimal elastic energy

Nonlinear functionals that appear as a product of two integrals are considered in the context of elastic curves of variable length. A technique is introduced that exploits the fact that one of the integrals has an integrand independent of the derivative of the unknown. Both the linear and the nonlinear cases are illustrated. By lengthening parameterized curves it is possible to reduce the elastic energy to zero. It is shown here that for graphs this is not the case. Specifically, there is a unique graph of minimal elastic energy among all graphs that have turned 90 degrees after traversing one unit.

Symmetrized curve-straightening

Abstract The ‘traditional’ curve-straightening flow is based on one of the standard Sobolev inner products and it is known to break certain symmetries of reflection. The purpose of this paper is to show that there are alternative Riemannian structures on the space of curves that yield flows that preserve symmetries. This feature comes at a price. In one symmetrizing metric the gradient vector fields are considerably more demanding to compute. In another symmetrizing metric smoothness is lost. This investigation will also explain the phenomena of ‘spinning’ as observed in several examples in the traditional flow. Three classes of alternative Riemannian structures are examined. The first class includes the traditional metric as a special case and is shown to never preserve both rotation symmetries and symmetries of reflection. The second class consists of a single metric corresponding to one of the standard Sobolev metrics, and is shown to preserve both types of symmetries. The third class also includes the traditional metric but it is shown that there is a unique different metric in this class, which preserves both types of symmetries. This particular metric generally yields smooth vector fields, which when evaluated at a smooth function do not give a smooth element of the corresponding tangent space. The third class is nevertheless ‘preferred’ since it has the distinction that it ‘respects’ the projection induced by the derivative operator onto the tangent bundle of the space of derivatives. The paper concludes with a number of graphical illustrations that show preserved symmetry and removal of spinning.

Curve-straightening in closed Euclidean submanifolds

The flow in the negative direction of the gradient vector field associated with the functional total squared (geodesic) curvature ∫k2ds is the so-called curvestraightening flow. This paper will consider spaces of closed curves in closed Euclidean submanifolds. It will define these spaces of curves as submanifolds of certain Hilbert manifolds representing all curves. The main result will then be to show the existence of a particular set of functionals defined on the entire Hilbert manifold which have the following four properties: 1. The directional derivatives of these functionals may be computed by solving an initial value problem for a system of ordinary differential equations. 2. By introducing a suitable Hilbert space basis for the Sobolev spaces used, the gradients may be effectively computed (but of course not explicitly computed, except in very special cases). 3. The gradients span the space normal to the tangent space of the space of closed curves. 4. Despite the fact that these gradients in general are not given explicitly it is nevertheless possible to compute the projection onto the tangent space to the space of closed curves. In particular we do this for the gradient of ∫k2ds. When all details are worked out this gives us an algorithm (which we supply) for finding critical points in the space of closed curves. It is not known if the trajectories actually always converge to critical points. If the functional is modified to include a multiple of the length so the functional becomes ∫k2+λds then the above convergence is known for λ>0. The motivating application for the curvestraightening flow is the possibility of using it to find (non-trivial) closed (periodic) geodesics. Note that if λ=0 then a closed geodesic is a global minimum. For any λ, geodesics are critical but there are also other critical points, the so-called elastic curves. The paper concludes by deriving the second variation formula for ∫k2+λds along closed geodesics. The quadratic functional associated with the second derivative is shown to be positive definite even for non-zero λ along some closed geodesics in some particular manifolds of interest.

Periodic Geodesics Generator

A method to generate periodic geodesics in arbitrary level surfaces is presented. The underlying algorithm resolves several technical complications posed by the constraints to stay in the surface and retain periodicity. The method exploits the “inverse” of the parallel transport equation and its “derivative.” This approach avoids most of the complications due to the intricate form of the geodesic curvature. The process flows any periodic curve in the surface along the negative gradient trajectory of the total squared geodesic curvature. The mathematical framework is that of an infinite-dimensional Riemannian manifold representing periodic curves of arbitrary length. The method is illustrated by an example in a sphere-like surface that is neither an ellipsoid nor a surface of revolution.

Discrete periodic geodesics in a surface

An alternative to the traditional curve-straightening flow on periodic curves in surfaces is introduced. The implementation of this flow produces periodic geodesics in minutes rather than hours. The flow is also simpler to initiate since its use of a penalty method permits initial curves that are not necessarily in the surface. Compact and noncompact examples are provided as well as examples with trivial and nontrivial free homotopy classes. The explicit curve-straightening flow on circles in Euclidean space is derived to help check the consistency of the implementations.